Science:Math Exam Resources/Courses/MATH103/April 2014/Question 09 (b)

MATH103 April 2014

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Question 09 (b)
Consider the iterated map $x_{n+1}=F(x_{n})$ where $F(x)={\frac {3}{2}}x(1-x)$ for $0\leq x\leq 1$. Sketch the cobweb corresponding to the sequence $x_{n}$ with $x_{0}=1/8$. No description

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Start by identifying the point $(x_{0},F(x_{0}))$ on the graph.

Hint 2
Next, use the line $y=x$ to find the point $(F(x_{0}),F(x_{0}))$.

Hint 3
Next, use this point $(F(x_{0}),F(x_{0}))$ to find $(F(x_{0}),F^{2}(x_{0}))$. Then repeat this procedure.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Cobweb The purpose of a cobweb plot is to determine graphically the sequence of points $x_{0},F(x_{0}),F(F(x_{0})),\ldots$ determined by iterating $F$ (for some initial condition $x_{0}$). Starting at $x_{0}$, our first goal is to determine $F(x_{0})$. To do so, begin by tracing a vertical line from $(x_{0},0)$ to $(x_{0},F(x_{0}))$. Once we have found $(x_{0},F(x_{0}))$ (see the first hint), we want to determine where $F(x_{0})$ lies on the $x$-axis (so that we can repeat this procedure). To do so, we begin by finding $(F(x_{0}),F(x_{0}))$. Since the $x$- and $y$-coordinates of this point agree with each other, it lies on the line $y=x$. Thus, we sketch the line $y=x$ and determine the point on this line with height $F(x_{0})$. To find the point on the line $y=F(x)$ of height $F(x_{0})$, take the point $(x_{0},F(x_{0}))$ found in Hint 1 and draw a horizontal line towards the line $y=x$. The intersection of these two lines is $(F(x_{0}),F(x_{0}))$. Once we have found $(F(x_{0}),F(x_{0}))$ (see Hint 3), we can get determine the point $F(x_{0})$ on the $x$-axis simply by drawing a vertical line from this point to the $x$-axis. We can now repeat this procedure (replacing $x_{0}$ by $F(x_{0})$, $F(x_{0})$ by $F(F(x_{0}))$, etc.).